Non-equilibrium Green's function (NEGF). To perform the NEGF calculation, the square-octagon lattice sample was divided into three parts – the left lead, the central part, and the right lead. The lead temperatures were kept at unequal \(T_L\)  (left) and \(T_R\)  (right) values to ensure a thermally non-equilibrium state. The magnon transport of the central part is what we are concerned with [1]}. The Hamiltonian for this part can be written as
\(\mathcal {H}=\sum _{\alpha , \beta =L, C, R} b_{\alpha }^{\dag } H_{\alpha \beta } b_{\beta }.\) 
